qualitative behavior of ion acoustic waves in nonextensive...

JURNAL FIZIK MALAYSIA Vol. 37, Issue 1, pp. 01102-01115 (2016) A. Saha

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Qualitative Behavior of Ion Acoustic Waves in Nonextensive Unmagnetized Plasmas

Asit Saha1,*, Tapash Saha1, Malay Kumar Ghorui2 and Prasanta Chatterjee3

1Department of Mathematics, Sikkim Manipal Institute of Technology, Sikkim Manipal University, Majitar, Rangpo, East-Sikkim 737136, India

2Department of Mathematics, B.B. College, Ushagram, Asansol 713303, India 3Department of Mathematics, Siksha-Bhavana, Visva-Bharati University, Santiniketan 731235, India

*[email protected]

(Received: 28 July 2016; published 17 December 2016)

Abstract. The qualitative behaviors of ion acoustic waves in unmagnetized plasma with q-nonextensive cool and hot electrons are investigated using the bifurcation theory of planar dynamical systems. Using a traveling wave transformation and initial conditions, basic equations are transformed to a planar dynamical system. Using numerical computations, all possible phase portraits of the dynamical system are presented. Corresponding to homoclinic and periodic orbits of the phase portraits, three analytical forms of solitary and periodic wave solutions are derived depending on nonextensive parameter , ratio of unperturbed densities of cold electrons and ions , temperature ratio of cold and hot electrons and speed of the traveling wave. Considering an external periodic perturbation, the quasiperiodic and chaotic behaviours of ion acoustic waves are presented. Depending upon different ranges of nonextensive parameter , the effect of is shown on quasiperiodic and chaotic behaviours of ion acoustic waves with fixed value of , and .

Keywords: Ion acoustic wave; unmagnetized plasma; solitary wave; bifurcation; chaos.

I. INTRODUCTION

The study of nonlinear wave phenomena in plasmas is one of the most important research areas of plasma physics. Formation and propagation of solitary wave, periodic wave and double layer are among the most interesting and challenging problems in space plasmas. Recently, considering an external perturbation, quasiperiodic and chaotic motions of nonlinear waves in plasmas have been received a great attention by many researchers [1]-[4]. Washimi and Taniuti [5] studied small amplitude nonlinear ion-acoustic solitary waves in a cold collisionless plasma in the framework of the Kortweg-deVries (KdV) equation. Considering a set of fluid equations, Sagdeev [6] investigated the properties of solitons in plasmas whose constituents are hot isothermal electrons and cold ions. Many researchers performed their extensive investigations on laboratory experiment [7]-[8] and theoretical analysis [9]-[11] for the existence and behaviors of ion-acoustic solitary waves in different plasmas. Later on, a number of works have been reported based on the propagations of solitary waves along the auroral magnetic field lines by several


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spacecraft missions: S3-3 [12], Viking [13], FREJA [14], POLAR [15]-[16] and FAST [17] satellites. Many authors [18]-[20] investigated nonlinear Solitary waves and double layers in plasmas containing multi-temperature electrons. Johnston and Epstein [21] studied the nonlinear ion-acoustic solitary waves in cold collisionless plasma applying the direct analysis of the field equations. They pointed out that a very small change in the initial condition breaks up the oscillatory behaviour of the solitary waves. Maitra and Roychoudhury [22] investigated dust-acoustic solitary waves applying the same analysis. Recently, Jain and Mishra [23] studied the large amplitude ion acoustic double layers in collisionless plasma whose constituents are isothermal positrons, warm adiabatic ions and multi-temperature electrons through the pseudo potential method. Very recently, Roy et al. [24] studied the head-on collisions and overtaking collisions of dust acoustic multi-solitons in dusty plasma with -nonextensive velocity distributed ions applying the Extended Poincare-Lighthill-Kuo (EPLK) method and using the results of Hirota’s direct method.

It is important to be noted that the Maxwell distribution is valid for the macroscopic ergodic equilibrium state and it may not be adequate to perform the long-range interactions in unmagnetized collision less plasma with non-equilibrium stationary state, which may exist depending upon a number of physical mechanisms (external force field present in natural space plasma environments, wave-particle interaction, and turbulence).

In order to model an electron distribution with nonextensive particles, one can use a nonextensive electron distribution function [25] given by

1 12

,

where denotes the electrostatic potential and the remaining variables or parameters obey their usual meaning. It is really important to note that is the special distribution which maximizes the Tsallis entropy and, thus, conforms to the laws of thermodynamics. In this case, the constant of normalization is given by

Γ

Γ for 1 1,

and

Γ

Γ for 1,

Here, the parameter stands for the strength of nonextensivity and Γ is the standard gamma function. It is important to note that for 1, the -distribution is unnormalizable. In the extensive limiting case → 1, the distribution reduces to the well-known Maxwell-Boltzmann


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velocity distribution. Integrating over all velocity space, we obtain the following nonextensive electron number density as:

1 1 .

Recently, Samanta et al. [26] studied nonlinear dust ion acoustic waves in the magnetized dusty plasma with q-nonextensive electrons on the framework of the KP equation using the bifurcation theory of planar dynamical systems for the first time. Samanta et al. [27] investigated bifurcations of nonlinear propagation of ion acoustic waves (IAWs) in a magnetized plasma. Saha and Chatterjee [28] extended the work [26] considering higher order nonlinearity in the framework of the MKP equation. Applying the same theory, Saha and Chatterjee addressed dust acoustic solitary and periodic waves [29]- [30] in dusty plasmas, ion acoustic solitary and periodic waves in e-p-i plasma [31] and ion acoustic solitary and periodic waves in a three component unmagnetized plasma [32] considering the terms involving electrostatic potential up to second degree in the Poisson equation. Saha et al. [1] investigated the dynamic behaviour of ion acoustic waves in electron-positron-ion magnetoplasmas with -distributed electrons and positrons on the framework of the Kadomtsev-Petviashili (KP) equation. Sahu et al. [2] studied the quasi periodic behaviour in quantum plasmas due to the presence of bohm potential. Very recently, Saha et al. [33] studied the dynamic features of dust acoustic waves in four components dusty plasma with non-thermal ions. Saha et al. [34] also studied bifurcations and chaotic behaviors of dust acoustic travelling waves in magnetoplasmas with nonthermal ions featuring Cairns-Tsallis distribution on the framework of the further modified Kadomtsev-Petviashili (FMKP) equation.

In this work, we consider the same model equations of the work [32] with q-nonextensive cold and hot electrons and extend our study. Considering the terms of electrostatic potential _ up to third degree in the Poisson equation, we obtain a dynamical system containing three equilibrium points. We present a new set of phase portraits with three equilibrium points. We derive three analytical forms for solitary and periodic waves corresponding to homoclinic orbits and periodic orbits in the phase portraits. Considering an external periodic perturbation, we study the quasiperiodic and chaotic behaviours of the perturbed system. The remaining part of the paper is organized as follows: In section II, we consider basic equations. In Section III, we obtain a planar dynamical system. Analytical wave solutions are derived in Section IV. We present quasiperiodic and chaotic behaviors of the perturbed system in section V and section VI is kept for conclusions.

II. BASIC EQUATIONS

In this work, we consider a three-component collisionless unmagnetized plasma which contains singly charged cold inertial fluid ions, and q-nonextensive cold and hot electrons. In equilibrium, the charge neutrality condition is , where , and are the


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unperturbed number densities of cold electron, hot electron and ion, respectively. The normalized basic equations are as follows:

0, (1)

, (2)

. (3)

The number densities of the q-nonextensive velocity distributed cold and hot electrons are given by

1 1 , (4)

1 1 1 , (5)

where , , and denote the number densities of cold electrons, hot electrons and ions, respectively, normalized by unperturbed number density . Here and denote the ion fluid velocity and electrostatic potential, respectively, normalized by the ion acoustic speed

/ / , and / , where is the electron charge and is the mass of ions. The time variable is normalized by the ion plasma period 4 / / and the space variable is normalized by the Debye length /4 / . Furthermore, we denote / , and

/ .

III. DYNAMICAL SYSTEM AND PHASE PLANE ANALYSIS In this section, we transform our model equations into a planar dynamical system. To do so, we introduce a new variable , where is the speed of the ion acoustic travelling wave. Substituting the new variable into equations (1) and (2) and using the initial conditions

0, 1and 0, we can express the ion number density as

. (6)

Substituting equations (4), (5) and (6) into equation (3) and considering the terms involving up to third degree, we have

, (7)

where 1 , 1 , and


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1 .

Then equation (7) is equivalent to the following planar dynamical system:

,

. (8)

Thus, the system (8) is a planar Hamiltonian system with Hamiltonian function:

, . (9)

The system (8) is a planar dynamical system with parameters , , and . It is clear that the phase orbits defined by the vector fields of system (8) will determine all travelling wave solutions of equation (7). A homoclinic orbit of equation (8) gives a solitary wave solution of the system. Similarly, a periodic orbit of equation (8) gives a periodic travelling wave solution of the system. In this study, the bifurcation theory of planar dynamical systems plays an important role [35]-[37].

We study the bifurcation set and phase portraits of the planar Hamiltonian system (8). Clearly, on the , phase plane, the abscissas of equilibrium points of system (8) are the zeros

of . Let , 0 be an equilibrium point of the dynamical system (8)

where 0. When 4 0, there exist three equilibrium points at

, 0 , , 0 and , 0 where 0, √

and√

. If

, 0 is the coefficient matrix of the linearized system of the travelling system (8) at an equilibrium point , 0 , then we get

, 0 . (10)

By the theory of planar dynamical systems [35]-[37], it is clear that the equilibrium point , 0 of the planar dynamical system (8) is a saddle point when 0 and the equilibrium

point , 0 of the planar dynamical system (8) is a centre when 0. Applying the systematic analysis of the physical parameters , , and , we have presented the phase portraits of the system (8) in figures 1-6.


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FIGURE 1. Phase portrait of equation (8) for 1.2, 0.3, 0.4 and 1.3.

FIGURE 2. Phase portrait of equation (8) for 0.9 and other parameters are same as FIGURE 1.

i) When 0, 0, 0, and 4 0, then the system (8) has three

equilibrium points at , 0 , , 0 and , 0 with 0 , where , 0 and , 0 are centres, and , 0 is a saddle point. There is a pair of homoclinic orbits at , 0 enclosing the centers at , 0 and , 0 (see figure 1).



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FIGURE 5. Phase portrait of equation (8) for 1.2, 2.03 and other parameters are same as FIGURE 1.

ii) When 0, 0, 0, 4 0and2 9 0, then the system (8) has three equilibrium points at , 0 , , 0 and , 0 with0, where , 0 and , 0 are centres, and , 0 is a saddle point. There is a pair of homoclinic orbits at , 0 enclosing the centres at , 0 and , 0 (see figure 2).

FIGURE 6. Phase portrait of equation (8) for 1.2, 1.9 and other parameters are same as FIGURE 1.


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iii) When 0, 0, 0, 4 0and2 9 0, then the system (8) has three equilibrium points at , 0 , , 0 and , 0 with 0 , where , 0 and , 0 are saddle point, and , 0 is a centre. There is a pair of homoclinic orbits at , 0 enclosing the centre at , 0 (see figure 3). iv) When 0, 0, 0, 4 0and2 9 0, then the system (8) has three equilibrium points at , 0 , , 0 and , 0 with 0 , where , 0 and , 0 are saddle point, and , 0 is a centre. There is a pair of homoclinic orbits at , 0 enclosing the centre at , 0 (see figure 4). v) When 0, 0, 0, 4 0and2 9 0, then the system (8) has three equilibrium points at , 0 , , 0 and , 0 with 0, where

, 0 and , 0 are saddle points, and , 0 is a centre. There is a pair of homoclinic orbits at , 0 enclosing the centre at , 0 (see figure 5). vi) When 0, 0, 0, 4 0and2 9 0, then the system (8) has three equilibrium points at , 0 , , 0 and , 0 with 0, where

, 0 is a saddle point, and , 0 and , 0 are centres. There is a pair of homoclinic orbits at , 0 enclosing the centres at , 0 and , 0 (see figure 6).

IV. ANALYTICAL SOLUTIONS In this section, we present solitary wave solutions and periodic wave solutions with the help of the dynamical system (8) and the Hamiltonian function (9). It should be noted that

Ω , is the Jacobian elliptic function [38] with modulo .

i) Corresponding to pair of homoclinic orbits at , 0 in figure 1, our system has both compressive and rarefactive solitary wave solutions:

. (11)

ii) Corresponding to family of periodic orbits about , 0 in figure 3, our system has a family of periodic wave solutions:

,

,, (12)


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with Ω and , where , , and are roots of

the equation 0, with , ∈ , 0 .

iii) Corresponding to homoclinic orbit at , 0 in figure 5, our system has solitary wave solution:

, (13)

where √ .

It should be noted that corresponding to homoclinic and periodic orbits of the other phase portraits (figures 2, 4 and 6) of the system (8), one can obtain the solitary and periodic wave solutions.

V. QUASIPERIODIC AND CHAOTIC BEHAVIORS

In this section, we will present the quasiperiodic and chaotic behaviors of the following perturbed system:

,

cos , (14)

where cos is an external periodic perturbation, is strength of the external perturbation and is the frequency. The difference between the system (8) and the system (14) is that only external periodic perturbation is added with the system (8).

It is important to note that the integrability of a system could be destroyed due to the effect of external periodic perturbations occurring in some real physical environments [39]-[41]. The type of the external periodic perturbation may vary depending upon different physical situations. A significant attention is recently received on the study of nonlinear evolution equations considering an external periodic perturbation as completely integrable nonlinear wave equation is unable to describe quasiperiodic or chaotic features. But, presence of an external periodic perturbation to a nonlinear integrable wave equation may lead to quasiperiodic or chaotic motions. In figures 7 and9, we have presented phase portraits of the perturbed dynamical system (14) for different values of (1.2 (see figure7), 0.9 (see figure9),) with fixed values of other


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parameters 0.3, 0.4, 1.3, 1.9 and 2.12. In figures 8 and10, we have plotted vs. for the perturbed system (14) with same values of parameters as figures 7 and 9, respectively. It is found that the perturbed system (14) represents chaotic behaviors for some special values of in the ranges 1 and0 1. In figure11, we have presented phase portrait of the perturbed dynamical system (14) for 1.2, 0.3, 0.4, 1.9,0.6 and 0.5. In figure 12, we have plotted vs. for the perturbed system (14) with same values of parameters as figure 11.

FIGURE 7. Phase portrait of equation (14) for 1.2, 0.3, 0.4, 1.3, 1.9 and

2.12 with initial condition 0.65, 0.193.

FIGURE 8. Plot of vs. with the same set of values of parameters as Figure 7.

Furthermore, the developed chaotic motions occur (see figures 7-12) and the solutions ignore the periodic motions and represent random sequences of uncorrelated oscillations. In figure13, we have presented phase portrait of the perturbed dynamical system (14) for 0.3, 0.3,

0.1, 1.9, 1.2 and 1.6. In figure 14, we have plotted vs. for the perturbed system 14 with same values of parameters as figure13. It is clear that the perturbed system (14) shows quasiperiodic behaviors for special value of in the range0 1. It is easily seen that the quasiperiodic and chaotic behaviors of ion acoustic waves are visible in the system (14) for some special values of in different regimes.


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VI. CONCLUSIONS We have addressed the qualitative behaviors of ion acoustic waves in three-component collisionless unmagnetized plasma with cold inertial ions, -nonextensive cold and hot electrons. Applying phase portrait analysis, we have established that our model has ion acoustic solitary


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and periodic wave solutions. We have presented three analytical forms of ion acoustic wave solutions depending on , , and . Considering an external periodic perturbation, quasiperiodic and chaotic behaviors of ion acoustic waves have been presented. Depending upon different regimes of the nonextensive parameter , we have shown the effect of on quasiperiodic and chaotic behaviors of ion acoustic waves with some particular values of other parameters , and It is found that the perturbed system has the quasiperiodic behaviors for some special values of in the range0 1, and chaotic behaviors in the ranges 0 1 and 1. The results of this study may be helpful to understand the qualitative behaviors of nonlinear ion acoustic waves in different plasma systems, where an enhanced tail of high energy particles compared to a Maxwellian exists.

Acknowledgements

The authors are thankful to the editor and reviewers for their valuable suggestions and comments which helped to improve the paper.

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